List Decoding Barnes-Wall Lattices

Abstract

The question of list decoding error-correcting codes over finite fields (under the Hamming metric) has been widely studied in recent years. Motivated by the similar discrete linear structure of linear codes and point lattices in R N , and their many shared applications across complexity theory, cryptography, and coding theory, we initiate the study of list decoding for lattices. Namely: for a lattice L ⊆ R N , given a target vector r ∈ R N and a distance parameter d, output the set of all lattice points w ∈ L that are within distance d of r.In this work we focus on combinatorial and algorithmic questions related to list decoding for the well-studied family of Barnes-Wall lattices. Our main contributions are twofold:1. We give tight (up to polynomials) combinatorial bounds on the worst-case list size, showing it to be polynomial in the lattice dimension for any error radius bounded away from the lattice's minimum distance (in the Euclidean norm).2. Building on the unique decoding algorithm of Micciancio and Nicolosi (ISIT '08), we give a listdecoding algorithm that runs in time polynomial in the lattice dimension and worst-case list size, for any error radius. Moreover, our algorithm is highly parallelizable, and with sufficiently many processors can run in parallel time only poly-logarithmic in the lattice dimension.In particular, our results imply a polynomial-time list-decoding algorithm for any error radius bounded away from the minimum distance, thus beating a typical barrier for natural error-correcting codes posed by the Johnson radius.

Practical Encoders and Decoders for Euclidean Codes from Barnes-Wall Lattices

“…Our goal is to construct efficient BW lattice codes of large block lengths which work with lowcomplexity encoders and decoders. In particular, efficient low-complexity decoders for complex BW lattices are readily available in [13], [15]. Therefore, if lattice codes from complex BW lattices are employed for communication over AWGN channels, then the decoders of [13], [15] can be used to recover information with low computational complexity.…”

“…In [13], the authors have proposed two low-complexity implementations of the bounded distance decoder for BW lattices, namely (i) the sequential bounded distance decoder, and (ii) the parallel bounded distance decoder. Inspired by the parallel decoder in [13], list decoders for BW lattices have been recently proposed in [15]. We note that the parallel decoders of [13] and [15] have low-complexity only when implemented on sufficiently large number of parallel processors.…”

“…Inspired by the parallel decoder in [13], list decoders for BW lattices have been recently proposed in [15]. We note that the parallel decoders of [13] and [15] have low-complexity only when implemented on sufficiently large number of parallel processors.…”

“…For the fully sequential decoder, the algorithm is assumed to be implemented on a single processor, whereas for the fully parallel decoder, the algorithm is assumed to be implemented on N 2 parallel processors. Inspired by the fully parallel implementation in [13], list decoders for BW lattices have been recently proposed in [15] where the list decoder outputs a list of BW lattice points within any given radius from the target vector. The complexity of the list decoder is shown to be polynomial in the dimension of the lattice, and polynomial in the list size, which is a function of the Euclidean radius.…”

“…In Fig. 3, the empirical values of P c are presented along with the bound in (15) for various values of SNR = 1 σ 2 . We point out that the bound is not tight due to the Chernoff-bound on each Q(·) function.…”